Machine Learning Force Fields

In recent years, the use of Machine Learning (ML) in computational chemistry has enabled numerous advances previously out of reach due to the computational complexity of traditional electronic-structure methods. One of the most promising applications is the construction of ML-based force fields (FFs), with the aim to narrow the gap between the accuracy of ab initio methods and the efficiency of classical FFs. The key idea is to learn the statistical relation between chemical structure and potential energy without relying on a preconceived notion of fixed chemical bonds or knowledge about the relevant interactions. Such universal ML approximations are in principle only limited by the quality and quantity of the reference data used to train them. This review gives an overview of applications of ML-FFs and the chemical insights that can be obtained from them. The core concepts underlying ML-FFs are described in detail and a step-by-step guide for constructing and testing them from scratch is given. The text concludes with a discussion of the challenges that remain to be overcome by the next generation of ML-FFs.

[1]  Janet E. Jones On the determination of molecular fields. —II. From the equation of state of a gas , 1924 .

[2]  Sergios Theodoridis,et al.  Machine Learning: A Bayesian and Optimization Perspective , 2015 .

[3]  Vijay S. Pande,et al.  Molecular graph convolutions: moving beyond fingerprints , 2016, Journal of Computer-Aided Molecular Design.

[4]  L. Verlet Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules , 1967 .

[5]  Frank Noé,et al.  Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics , 2017, The Journal of chemical physics.

[6]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[7]  Michele Ceriotti,et al.  Nuclear quantum effects enter the mainstream , 2018, 1803.01037.

[8]  V. Barone,et al.  Toward reliable density functional methods without adjustable parameters: The PBE0 model , 1999 .

[9]  Bertrand Guillot,et al.  A reappraisal of what we have learnt during three decades of computer simulations on water , 2002 .

[10]  Frank Noé,et al.  Efficient multi-objective molecular optimization in a continuous latent space† †Electronic supplementary information (ESI) available: Details of the desirability scaling functions, high resolution figures and detailed results of the GuacaMol benchmark. See DOI: 10.1039/c9sc01928f , 2019, Chemical science.

[11]  Heekuck Oh,et al.  Neural Networks for Pattern Recognition , 1993, Adv. Comput..

[12]  Frank Jensen,et al.  Force field modeling of conformational energies: Importance of multipole moments and intramolecular polarization , 2007 .

[13]  Markus Meuwly,et al.  Minimal distributed charges: Multipolar quality at the cost of point charge electrostatics. , 2017, The Journal of chemical physics.

[14]  Jörg Behler,et al.  A neural network potential-energy surface for the water dimer based on environment-dependent atomic energies and charges. , 2012, The Journal of chemical physics.

[15]  David W Toth,et al.  The TensorMol-0.1 model chemistry: a neural network augmented with long-range physics , 2017, Chemical science.

[16]  Noam Bernstein,et al.  Machine learning unifies the modeling of materials and molecules , 2017, Science Advances.

[17]  Adrian E. Roitberg,et al.  Less is more: sampling chemical space with active learning , 2018, The Journal of chemical physics.

[18]  Alexander Binder,et al.  Unmasking Clever Hans predictors and assessing what machines really learn , 2019, Nature Communications.

[19]  Joachim M. Buhmann,et al.  On Relevant Dimensions in Kernel Feature Spaces , 2008, J. Mach. Learn. Res..

[20]  Klaus-Robert Müller,et al.  Kernel Analysis of Deep Networks , 2011, J. Mach. Learn. Res..

[21]  Wei-Hai Fang,et al.  Deep Learning for Nonadiabatic Excited-State Dynamics. , 2018, The journal of physical chemistry letters.

[22]  Risi Kondor,et al.  Diffusion kernels on graphs and other discrete structures , 2002, ICML 2002.

[23]  Frank Noé,et al.  Deep-neural-network solution of the electronic Schrödinger equation , 2020, Nature Chemistry.

[24]  Vasily V. Bulatov,et al.  Quantum effects on dislocation motion from ring-polymer molecular dynamics , 2017, npj Computational Materials.

[25]  Michael Rabadi,et al.  Kernel Methods for Machine Learning , 2015 .

[26]  Noam Bernstein,et al.  Machine Learning a General-Purpose Interatomic Potential for Silicon , 2018, Physical Review X.

[27]  Gábor Csányi,et al.  Comparing molecules and solids across structural and alchemical space. , 2015, Physical chemistry chemical physics : PCCP.

[28]  Alexandre Tkatchenko,et al.  Accurate Many-Body Repulsive Potentials for Density-Functional Tight Binding from Deep Tensor Neural Networks. , 2020, The journal of physical chemistry letters.

[29]  Markus Meuwly,et al.  Multisurface Adiabatic Reactive Molecular Dynamics. , 2014, Journal of chemical theory and computation.

[30]  M. Mezei,et al.  Molecular docking: a powerful approach for structure-based drug discovery. , 2011, Current computer-aided drug design.

[31]  Jörg Behler,et al.  Nuclear Quantum Effects in Sodium Hydroxide Solutions from Neural Network Molecular Dynamics Simulations. , 2018, The journal of physical chemistry. B.

[32]  R. Kondor,et al.  On representing chemical environments , 2012, 1209.3140.

[33]  V. Raykar,et al.  Fast large scale Gaussian process regression using approximate matrix-vector products , 2006 .

[34]  Jan Hermann,et al.  First-Principles Models for van der Waals Interactions in Molecules and Materials: Concepts, Theory, and Applications. , 2017, Chemical reviews.

[35]  Lu-Ming Duan,et al.  Machine learning meets quantum physics , 2019, Physics Today.

[36]  Hervé Abdi,et al.  A NEURAL NETWORK PRIMER , 1994 .

[37]  Jaehoon Lee,et al.  Deep Neural Networks as Gaussian Processes , 2017, ICLR.

[38]  Alireza Khorshidi,et al.  Amp: A modular approach to machine learning in atomistic simulations , 2016, Comput. Phys. Commun..

[39]  Bin Liu,et al.  Thermal Expansion of Single Wall Carbon Nanotubes , 2004 .

[40]  Benjamin Recht,et al.  Random Features for Large-Scale Kernel Machines , 2007, NIPS.

[41]  Ohad Shamir,et al.  The Power of Depth for Feedforward Neural Networks , 2015, COLT.

[42]  Kurt Hornik,et al.  Approximation capabilities of multilayer feedforward networks , 1991, Neural Networks.

[43]  Sergei Manzhos,et al.  A random-sampling high dimensional model representation neural network for building potential energy surfaces. , 2006, The Journal of chemical physics.

[44]  F. L. Hirshfeld Bonded-atom fragments for describing molecular charge densities , 1977 .

[45]  A. Tkatchenko,et al.  Modeling quantum nuclei with perturbed path integral molecular dynamics† †Electronic supplementary information (ESI) available: Heat capacity estimator and first and second-order cumulant expansions of the TI approach. See DOI: 10.1039/c5sc03443d , 2015, Chemical science.

[46]  Frank Noé,et al.  Equivariant Flows: exact likelihood generative learning for symmetric densities , 2020, ICML.

[47]  Jürgen Schmidhuber,et al.  Deep learning in neural networks: An overview , 2014, Neural Networks.

[48]  Peter G. Wolynes,et al.  Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids , 1981 .

[49]  Charles A. Micchelli,et al.  When is there a representer theorem? Vector versus matrix regularizers , 2008, J. Mach. Learn. Res..

[50]  Thomas Spura,et al.  "On-the-fly" coupled cluster path-integral molecular dynamics: impact of nuclear quantum effects on the protonated water dimer. , 2015, Physical chemistry chemical physics : PCCP.

[51]  Michael Gastegger,et al.  Machine learning molecular dynamics for the simulation of infrared spectra† †Electronic supplementary information (ESI) available. See DOI: 10.1039/c7sc02267k , 2017, Chemical science.

[52]  J S Smith,et al.  ANI-1: an extensible neural network potential with DFT accuracy at force field computational cost , 2016, Chemical science.

[53]  Detlef Hommel,et al.  Thermal expansion of bulk and homoepitaxial GaN , 2000 .

[54]  Paul K. Weiner,et al.  Ground states of molecules , 1972 .

[55]  Steven D. Brown,et al.  Neural network models of potential energy surfaces , 1995 .

[56]  Anders S. Christensen,et al.  Alchemical and structural distribution based representation for universal quantum machine learning. , 2017, The Journal of chemical physics.

[57]  Klaus Schulten,et al.  Mature HIV-1 capsid structure by cryo-electron microscopy and all-atom molecular dynamics , 2013, Nature.

[58]  C.E. Shannon,et al.  Communication in the Presence of Noise , 1949, Proceedings of the IRE.

[59]  Kristof T. Schütt,et al.  A deep neural network for molecular wave functions in quasi-atomic minimal basis representation. , 2020, The Journal of chemical physics.

[60]  A. Tkatchenko,et al.  Accurate molecular van der Waals interactions from ground-state electron density and free-atom reference data. , 2009, Physical review letters.

[61]  Klaus-Robert Müller,et al.  sGDML: Constructing accurate and data efficient molecular force fields using machine learning , 2018, Comput. Phys. Commun..

[62]  J. van Leeuwen,et al.  Neural Networks: Tricks of the Trade , 2002, Lecture Notes in Computer Science.

[63]  H. Sebastian Seung,et al.  Query by committee , 1992, COLT '92.

[64]  Julia Westermayr,et al.  Machine learning enables long time scale molecular photodynamics simulations , 2018, Chemical science.

[65]  Klaus-Robert Müller,et al.  Covariate Shift Adaptation by Importance Weighted Cross Validation , 2007, J. Mach. Learn. Res..

[66]  Klaus-Robert Müller,et al.  Exploring chemical compound space with quantum-based machine learning , 2020, Nature Reviews Chemistry.

[67]  Stéphane Mallat,et al.  Wavelet Scattering Regression of Quantum Chemical Energies , 2016, Multiscale Model. Simul..

[68]  M Gastegger,et al.  wACSF-Weighted atom-centered symmetry functions as descriptors in machine learning potentials. , 2017, The Journal of chemical physics.

[69]  Markus Meuwly,et al.  PhysNet: A Neural Network for Predicting Energies, Forces, Dipole Moments, and Partial Charges. , 2019, Journal of chemical theory and computation.

[70]  Klaus-Robert Müller,et al.  Introduction to machine learning for brain imaging , 2011, NeuroImage.

[71]  Arieh Warshel,et al.  An empirical valence bond approach for comparing reactions in solutions and in enzymes , 1980 .

[72]  Zhenwei Li,et al.  Molecular dynamics with on-the-fly machine learning of quantum-mechanical forces. , 2015, Physical review letters.

[73]  Andrea Grisafi,et al.  Using Gaussian process regression to simulate the vibrational Raman spectra of molecular crystals , 2019, New Journal of Physics.

[74]  Stewart A. Adcock,et al.  Molecular dynamics: survey of methods for simulating the activity of proteins. , 2006, Chemical reviews.

[75]  M. Plesset,et al.  Note on an Approximation Treatment for Many-Electron Systems , 1934 .

[76]  E Weinan,et al.  End-to-end Symmetry Preserving Inter-atomic Potential Energy Model for Finite and Extended Systems , 2018, NeurIPS.

[77]  T. Morawietz,et al.  How van der Waals interactions determine the unique properties of water , 2016, Proceedings of the National Academy of Sciences.

[78]  Felix A Faber,et al.  Machine Learning Energies of 2 Million Elpasolite (ABC_{2}D_{6}) Crystals. , 2015, Physical review letters.

[79]  Gábor Csányi,et al.  Gaussian approximation potentials: A brief tutorial introduction , 2015, 1502.01366.

[80]  Klaus-Robert Müller,et al.  Quantum-chemical insights from interpretable atomistic neural networks , 2018, Explainable AI.

[81]  R. Car,et al.  Raman spectrum and polarizability of liquid water from deep neural networks. , 2020, Physical chemistry chemical physics : PCCP.

[82]  Sergei Manzhos,et al.  Using redundant coordinates to represent potential energy surfaces with lower-dimensional functions. , 2007, The Journal of chemical physics.

[83]  Jun Zhang,et al.  Targeted Adversarial Learning Optimized Sampling. , 2019, The journal of physical chemistry letters.

[84]  T. H. Dunning Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen , 1989 .

[85]  M C Payne,et al.  "Learn on the fly": a hybrid classical and quantum-mechanical molecular dynamics simulation. , 2004, Physical review letters.

[86]  F. Noé,et al.  Dynamic properties of force fields. , 2015, The Journal of chemical physics.

[87]  Stephan Günnemann,et al.  Directional Message Passing for Molecular Graphs , 2020, ICLR.

[88]  Markus Meuwly,et al.  Reactive atomistic simulations of Diels-Alder reactions: The importance of molecular rotations. , 2019, The Journal of chemical physics.

[89]  Jos'e Miguel Hern'andez-Lobato,et al.  Reinforcement Learning for Molecular Design Guided by Quantum Mechanics , 2020, ICML.

[90]  K-R Müller,et al.  SchNetPack: A Deep Learning Toolbox For Atomistic Systems. , 2018, Journal of chemical theory and computation.

[91]  Jörg Behler,et al.  Constructing high‐dimensional neural network potentials: A tutorial review , 2015 .

[92]  Elie Bienenstock,et al.  Neural Networks and the Bias/Variance Dilemma , 1992, Neural Computation.

[93]  Frank Noé,et al.  Machine learning for protein folding and dynamics. , 2019, Current opinion in structural biology.

[94]  Simon Haykin,et al.  Neural Networks and Learning Machines , 2010 .

[95]  Hillary Sanders,et al.  GARBAGE IN, GARBAGE OUT: HOW PURPORTEDLY GREAT ML MODELS CAN BE SCREWED UP BY BAD DATA , 2017 .

[96]  Mark N. Gibbs,et al.  Combining ab initio computations, neural networks, and diffusion Monte Carlo: An efficient method to treat weakly bound molecules , 1996 .

[97]  Feliks Nüske,et al.  Sparse learning of stochastic dynamical equations. , 2017, The Journal of chemical physics.

[98]  Kristof T. Schütt,et al.  How to represent crystal structures for machine learning: Towards fast prediction of electronic properties , 2013, 1307.1266.

[99]  Li Li,et al.  Bypassing the Kohn-Sham equations with machine learning , 2016, Nature Communications.

[100]  R. Car,et al.  Free energy of proton transfer at the water–TiO2 interface from ab initio deep potential molecular dynamics† , 2020, Chemical science.

[101]  Mariana Rossi,et al.  Elucidating the Nuclear Quantum Dynamics of Intramolecular Double Hydrogen Transfer in Porphycene , 2018, Journal of the American Chemical Society.

[102]  Julia Westermayr,et al.  Combining SchNet and SHARC: The SchNarc Machine Learning Approach for Excited-State Dynamics , 2020, The journal of physical chemistry letters.

[103]  Andrea Grisafi,et al.  Symmetry-Adapted Machine Learning for Tensorial Properties of Atomistic Systems. , 2017, Physical review letters.

[104]  Markus Meuwly,et al.  Reactive dynamics and spectroscopy of hydrogen transfer from neural network-based reactive potential energy surfaces , 2019 .

[105]  T. Morawietz,et al.  A density-functional theory-based neural network potential for water clusters including van der Waals corrections. , 2013, The journal of physical chemistry. A.

[106]  Lorenzo Rosasco,et al.  On Invariance and Selectivity in Representation Learning , 2015, ArXiv.

[107]  Jonathan T. Barron,et al.  A General and Adaptive Robust Loss Function , 2017, 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).

[108]  Ah Chung Tsoi,et al.  The Graph Neural Network Model , 2009, IEEE Transactions on Neural Networks.

[109]  Steven Vandenbrande,et al.  i-PI 2.0: A universal force engine for advanced molecular simulations , 2018, Comput. Phys. Commun..

[110]  Gerbrand Ceder,et al.  Efficient and accurate machine-learning interpolation of atomic energies in compositions with many species , 2017, 1706.06293.

[111]  Yoshua Bengio,et al.  Random Search for Hyper-Parameter Optimization , 2012, J. Mach. Learn. Res..

[112]  K. Müller,et al.  Quantum chemical accuracy from density functional approximations via machine learning , 2020, Nature Communications.

[113]  Michael Gastegger,et al.  Generating equilibrium molecules with deep neural networks , 2018, ArXiv.

[114]  Klaus-Robert Müller,et al.  Finding Density Functionals with Machine Learning , 2011, Physical review letters.

[115]  Richard E. Turner,et al.  Gaussian Process Behaviour in Wide Deep Neural Networks , 2018, ICLR.

[116]  O. Anatole von Lilienfeld,et al.  The "DNA" of chemistry: Scalable quantum machine learning with "amons" , 2017, 1707.04146.

[117]  Henry S. Rzepa,et al.  Ground states of molecules: Part XLII. Vibrational frequencies of isotopically-substituted molecules calculated using MINDO/3 force constants , 1977 .

[118]  Michael Gastegger,et al.  Symmetry-adapted generation of 3d point sets for the targeted discovery of molecules , 2019, NeurIPS.

[119]  Matthias Rupp,et al.  Unified representation of molecules and crystals for machine learning , 2017, Mach. Learn. Sci. Technol..

[120]  Neha Agnihotri,et al.  Computational studies of charge transfer in organic solar photovoltaic cells: A review , 2014 .

[121]  Lorenzo Rosasco,et al.  FALKON: An Optimal Large Scale Kernel Method , 2017, NIPS.

[122]  Harold A. Scheraga,et al.  Description of the potential energy surface of the water dimer with an artificial neural network , 1997 .

[123]  H. Lischka,et al.  Multiconfiguration self-consistent field and multireference configuration interaction methods and applications. , 2012, Chemical reviews.

[124]  Michael Walter,et al.  The atomic simulation environment-a Python library for working with atoms. , 2017, Journal of physics. Condensed matter : an Institute of Physics journal.

[125]  J. J. Soares Neto,et al.  The fitting of potential energy surfaces using neural networks. Application to the study of the photodissociation processes , 1998 .

[126]  Walter Thiel,et al.  QM/MM methods for biomolecular systems. , 2009, Angewandte Chemie.

[127]  R. Kondor,et al.  Gaussian approximation potentials: the accuracy of quantum mechanics, without the electrons. , 2009, Physical review letters.

[128]  Nils M. Kriege,et al.  On Valid Optimal Assignment Kernels and Applications to Graph Classification , 2016, NIPS.

[129]  Richard D Wilkinson,et al.  Interpolation of intermolecular potentials using Gaussian processes. , 2017, The Journal of chemical physics.

[130]  Volker L. Deringer,et al.  Machine learning based interatomic potential for amorphous carbon , 2016, 1611.03277.

[131]  Hao Wu,et al.  Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning , 2018, Science.

[132]  A. V. Duin,et al.  ReaxFF: A Reactive Force Field for Hydrocarbons , 2001 .

[133]  Alexander D. MacKerell,et al.  Force Field for Peptides and Proteins based on the Classical Drude Oscillator. , 2013, Journal of chemical theory and computation.

[134]  Pengyu Y. Ren,et al.  The Polarizable Atomic Multipole-based AMOEBA Force Field for Proteins. , 2013, Journal of chemical theory and computation.

[135]  Samuel S. Schoenholz,et al.  Neural Message Passing for Quantum Chemistry , 2017, ICML.

[136]  Michele Parrinello,et al.  Generalized neural-network representation of high-dimensional potential-energy surfaces. , 2007, Physical review letters.

[137]  Bernhard Schölkopf,et al.  Kernel Principal Component Analysis , 1997, ICANN.

[138]  Frank Noé,et al.  Machine Learning of Coarse-Grained Molecular Dynamics Force Fields , 2018, ACS central science.

[139]  M. A. González,et al.  Force fields and molecular dynamics simulations , 2011 .

[140]  Thomas E Markland,et al.  Proton Network Flexibility Enables Robustness and Large Electric Fields in the Ketosteroid Isomerase Active Site. , 2017, The journal of physical chemistry. B.

[141]  Bernhard E. Boser,et al.  A training algorithm for optimal margin classifiers , 1992, COLT '92.

[142]  Gunnar Rätsch,et al.  Predicting Time Series with Support Vector Machines , 1997, ICANN.

[143]  A. Berlinet,et al.  Reproducing kernel Hilbert spaces in probability and statistics , 2004 .

[144]  Justin S. Smith,et al.  Hierarchical modeling of molecular energies using a deep neural network. , 2017, The Journal of chemical physics.

[145]  Carl E. Rasmussen,et al.  A Unifying View of Sparse Approximate Gaussian Process Regression , 2005, J. Mach. Learn. Res..

[146]  H. D. Merchant,et al.  Equations of state and thermal expansion of alkali halides , 1973 .

[147]  B. Berne,et al.  Efficient molecular dynamics and hybrid Monte Carlo algorithms for path integrals , 1993 .

[148]  Geoffrey J. Gordon,et al.  A Density Functional Tight Binding Layer for Deep Learning of Chemical Hamiltonians. , 2018, Journal of chemical theory and computation.

[149]  Frank Noé,et al.  Machine learning for molecular simulation , 2019, Annual review of physical chemistry.

[150]  Pavlo O. Dral,et al.  Quantum chemistry structures and properties of 134 kilo molecules , 2014, Scientific Data.

[151]  Kwang S. Kim,et al.  Theory and applications of computational chemistry : the first forty years , 2005 .

[152]  Matthias Troyer,et al.  Solving the quantum many-body problem with artificial neural networks , 2016, Science.

[153]  Frank Noé,et al.  Kernel methods for detecting coherent structures in dynamical data. , 2019, Chaos.

[154]  O. Anatole von Lilienfeld,et al.  On the role of gradients for machine learning of molecular energies and forces , 2020, Mach. Learn. Sci. Technol..

[155]  Allan,et al.  Molecular dynamics and ab initio total energy calculations. , 1986, Physical review letters.

[156]  Hao Wu,et al.  Deep Generative Markov State Models , 2018, NeurIPS.

[157]  P. Dirac Quantum Mechanics of Many-Electron Systems , 1929 .

[158]  Klaus-Robert Müller,et al.  Machine learning of accurate energy-conserving molecular force fields , 2016, Science Advances.

[159]  K-R Müller,et al.  SchNet - A deep learning architecture for molecules and materials. , 2017, The Journal of chemical physics.

[160]  E. Paquet,et al.  Molecular Dynamics, Monte Carlo Simulations, and Langevin Dynamics: A Computational Review , 2015, BioMed research international.

[161]  Anders S. Christensen,et al.  Operators in quantum machine learning: Response properties in chemical space. , 2018, The Journal of chemical physics.

[162]  Teuvo Kohonen,et al.  An introduction to neural computing , 1988, Neural Networks.

[163]  Cecilia Clementi,et al.  Learning Effective Molecular Models from Experimental Observables. , 2018, Journal of chemical theory and computation.

[164]  Michael A. Osborne,et al.  Preconditioning Kernel Matrices , 2016, ICML.

[165]  Geoffrey E. Hinton,et al.  Rectified Linear Units Improve Restricted Boltzmann Machines , 2010, ICML.

[166]  W S McCulloch,et al.  A logical calculus of the ideas immanent in nervous activity , 1990, The Philosophy of Artificial Intelligence.

[167]  Gunnar Rätsch,et al.  An introduction to kernel-based learning algorithms , 2001, IEEE Trans. Neural Networks.

[168]  George Cybenko,et al.  Approximation by superpositions of a sigmoidal function , 1989, Math. Control. Signals Syst..

[169]  Felix A Faber,et al.  Crystal structure representations for machine learning models of formation energies , 2015, 1503.07406.

[170]  Markus Meuwly,et al.  A reactive, scalable, and transferable model for molecular energies from a neural network approach based on local information. , 2018, The Journal of chemical physics.

[171]  Hao Wu,et al.  VAMPnets for deep learning of molecular kinetics , 2017, Nature Communications.

[172]  S. Plimpton,et al.  Short-Range Molecular Dynamics , 1995 .

[173]  Anthony Widjaja,et al.  Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond , 2003, IEEE Transactions on Neural Networks.

[174]  Bing Huang,et al.  Quantum machine learning using atom-in-molecule-based fragments selected on the fly , 2017, Nature Chemistry.

[175]  Anders S Christensen,et al.  FCHL revisited: Faster and more accurate quantum machine learning. , 2020, The Journal of chemical physics.

[176]  Zoubin Ghahramani,et al.  Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning , 2015, ICML.

[177]  Gunnar Rätsch,et al.  Input space versus feature space in kernel-based methods , 1999, IEEE Trans. Neural Networks.

[178]  K. Müller,et al.  Towards exact molecular dynamics simulations with machine-learned force fields , 2018, Nature Communications.

[179]  Risi Kondor,et al.  Cormorant: Covariant Molecular Neural Networks , 2019, NeurIPS.

[180]  Feliks Nüske,et al.  Coarse-graining molecular systems by spectral matching. , 2019, The Journal of chemical physics.

[181]  H. Hellmann,et al.  Einführung in die Quantenchemie , 2015 .

[182]  Michele Parrinello,et al.  Silicon Liquid Structure and Crystal Nucleation from Ab Initio Deep Metadynamics. , 2018, Physical review letters.

[183]  Zoubin Ghahramani,et al.  Sparse Gaussian Processes using Pseudo-inputs , 2005, NIPS.

[184]  Frank Noé,et al.  Generating valid Euclidean distance matrices , 2019, ArXiv.

[185]  D. Costarelli,et al.  Constructive Approximation by Superposition of Sigmoidal Functions , 2013 .

[186]  Burke,et al.  Generalized Gradient Approximation Made Simple. , 1996, Physical review letters.

[187]  Hua Guo,et al.  Comprehensive Investigations of the Cl + CH 3 OH → HCl + CH 3 O/CH 2 OH Reaction: Validation of Experiment and Dynamic Insights , 2020 .

[188]  Matthias Scheffler,et al.  Ab initio molecular simulations with numeric atom-centered orbitals , 2009, Comput. Phys. Commun..

[189]  Markus Meuwly,et al.  Sampling reactive regions in phase space by following the minimum dynamic path. , 2019, The Journal of chemical physics.

[190]  Vladimir N. Vapnik,et al.  The Nature of Statistical Learning Theory , 2000, Statistics for Engineering and Information Science.

[191]  Max Welling,et al.  3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data , 2018, NeurIPS.

[192]  E Weinan,et al.  Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics , 2017, Physical review letters.

[193]  S. Grimme,et al.  A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. , 2010, The Journal of chemical physics.

[194]  Bernhard Schölkopf,et al.  The connection between regularization operators and support vector kernels , 1998, Neural Networks.

[195]  Thomas Spura,et al.  Correction: "On-the-fly" coupled cluster path-integral molecular dynamics: impact of nuclear quantum effects on the protonated water dimer. , 2015, Physical chemistry chemical physics : PCCP.

[196]  Heather J. Kulik,et al.  How Large Should the QM Region Be in QM/MM Calculations? The Case of Catechol O-Methyltransferase , 2015, The journal of physical chemistry. B.

[197]  Kevin P. Murphy,et al.  Machine learning - a probabilistic perspective , 2012, Adaptive computation and machine learning series.

[198]  Sameer Varma,et al.  Machine learning approaches to evaluate correlation patterns in allosteric signaling: A case study of the PDZ2 domain. , 2018, The Journal of chemical physics.

[199]  D. Ruppert The Elements of Statistical Learning: Data Mining, Inference, and Prediction , 2004 .

[200]  Alexandre Tkatchenko,et al.  Quantum tunneling of thermal protons through pristine graphene. , 2016, The Journal of chemical physics.

[201]  O. A. von Lilienfeld,et al.  Communication: Understanding molecular representations in machine learning: The role of uniqueness and target similarity. , 2016, The Journal of chemical physics.

[202]  Wei Chen,et al.  Nonlinear Discovery of Slow Molecular Modes using Hierarchical Dynamics Encoders , 2019, The Journal of chemical physics.

[203]  P. Wormer,et al.  Theory and Applications of Computational Chemistry The First Forty Years , 2005 .

[204]  Jonathan Schaffer,et al.  What Not to Multiply Without Necessity , 2015 .

[205]  Matthias W. Seeger,et al.  Using the Nyström Method to Speed Up Kernel Machines , 2000, NIPS.

[206]  Jean-Philippe Vert,et al.  The optimal assignment kernel is not positive definite , 2008, ArXiv.

[207]  Bernhard Schölkopf,et al.  Nonlinear Component Analysis as a Kernel Eigenvalue Problem , 1998, Neural Computation.

[208]  San Cristóbal Mateo,et al.  The Lack of A Priori Distinctions Between Learning Algorithms , 1996 .

[209]  R. Friesner Ab initio quantum chemistry: methodology and applications. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[210]  Y. Takane,et al.  Generalized Inverse Matrices , 2011 .

[211]  Sebastian Ruder,et al.  An overview of gradient descent optimization algorithms , 2016, Vestnik komp'iuternykh i informatsionnykh tekhnologii.

[212]  Bin Jiang,et al.  Embedded Atom Neural Network Potentials: Efficient and Accurate Machine Learning with a Physically Inspired Representation. , 2019, The journal of physical chemistry letters.

[213]  Shinji Umeyama,et al.  An Eigendecomposition Approach to Weighted Graph Matching Problems , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

[214]  J. Behler Atom-centered symmetry functions for constructing high-dimensional neural network potentials. , 2011, The Journal of chemical physics.

[215]  Marwin H. S. Segler,et al.  Machine learning the ropes: principles, applications and directions in synthetic chemistry. , 2020, Chemical Society reviews.